Properties

Label 2.1.7.6a1.1-1.2.8a
Base 2.1.7.6a1.1
Degree \(2\)
e \(2\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{2} + \left(b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5} + a_{7} \pi^{4}\right) x + c_{14} \pi^{8} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.7.6a1.1
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[2]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $8$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $(C_7:C_3) \times C_2$ (show 2), $F_8:C_6$ (show 8), $C_2\wr C_7:C_3$ (show 6)
Hidden Artin slopes: $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ (show 4), $[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ (show 4), $[\frac{8}{7},\frac{8}{7},\frac{8}{7}]^{3}$ (show 2), $[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ (show 2), $[\ ]^{3}$ (show 2), $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ (show 2)
Indices of inseparability: $[7,0]$
Associated inertia: $[3,1]$
Jump Set: $[7,14]$ (show 1), $[7,23]$ (show 8), $[7,25]$ (show 4), $[7,27]$ (show 2), $[7,28]$ (show 1)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.1.14.20a1.1 $x^{14} + 2 x^{7} + 2$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[2]_{7}^{3}$ $[1]_{7}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 14]$
2.1.14.20a1.2 $x^{14} + 2 x^{7} + 6$ $(C_7:C_3) \times C_2$ (as 14T5) $42$ $2$ $[2]_{7}^{3}$ $[1]_{7}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[7, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 28]$
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